L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 4·7-s − 8-s + 9-s + 6·11-s + 2·12-s − 2·13-s + 4·14-s + 16-s − 17-s − 18-s − 8·21-s − 6·22-s − 2·24-s − 5·25-s + 2·26-s − 4·27-s − 4·28-s + 4·31-s − 32-s + 12·33-s + 34-s + 36-s + 4·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.74·21-s − 1.27·22-s − 0.408·24-s − 25-s + 0.392·26-s − 0.769·27-s − 0.755·28-s + 0.718·31-s − 0.176·32-s + 2.08·33-s + 0.171·34-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.710684355\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.710684355\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.49297642827790, −15.59701261222785, −15.39957496865513, −14.59229133657598, −14.20721122146874, −13.54751282503082, −13.06122787928628, −12.22246497714437, −11.86326758752552, −11.18175634901061, −10.19142998858016, −9.732669058602829, −9.365288500086130, −8.846226352027827, −8.369548906816165, −7.399864759355155, −7.098172718852221, −6.163391057224065, −5.995804647017067, −4.507091358111067, −3.767322815035053, −3.254464349437996, −2.528539852222390, −1.762334388164159, −0.6075611584413628,
0.6075611584413628, 1.762334388164159, 2.528539852222390, 3.254464349437996, 3.767322815035053, 4.507091358111067, 5.995804647017067, 6.163391057224065, 7.098172718852221, 7.399864759355155, 8.369548906816165, 8.846226352027827, 9.365288500086130, 9.732669058602829, 10.19142998858016, 11.18175634901061, 11.86326758752552, 12.22246497714437, 13.06122787928628, 13.54751282503082, 14.20721122146874, 14.59229133657598, 15.39957496865513, 15.59701261222785, 16.49297642827790